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  • More on HOD-supercompactness
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-08
    Arthur W. Apter; Shoshana Friedman; Gunter Fuchs

    We explore Woodin's Universality Theorem and consider to what extent large cardinal properties are transferred into HOD (and other inner models). We also separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact. For example, we produce a model where a proper class of supercompact cardinals are not HOD-supercompact but are supercompact in HOD. Additionally we introduce

  • Intuitionistic fixed point logic
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-09
    Ulrich Berger; Hideki Tsuiki

    We study the system IFP of intuitionistic fixed point logic, an extension of intuitionistic first-order logic by strictly positive inductive and coinductive definitions. We define a realizability interpretation of IFP and use it to extract computational content from proofs about abstract structures specified by arbitrary classically true disjunction free formulas. The interpretation is shown to be

  • Unbounded towers and products
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-08
    Piotr Szewczak; Magdalena Włudecka

    We investigate products of sets of reals with combinatorial covering properties. A topological space satisfies S1(Γ,Γ) if for each sequence of point-cofinite open covers of the space, one can pick one element from each cover and obtain a point-cofinite cover of the space. We prove that, if there is an unbounded tower, then there is a nontrivial set of reals satisfying S1(Γ,Γ) in all finite powers.

  • A microscopic approach to Souslin-tree construction, Part II
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-13
    Ari Meir Brodsky; Assaf Rinot

    In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ⋄-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of ⋄. In particular, we obtain a new weak sufficient

  • Indestructibility of ideals and MAD families
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-13
    David Chodounský; Osvaldo Guzmán

    In this survey paper we collect several known results on destroying tall ideals on countable sets and maximal almost disjoint families with forcing. In most cases we provide streamlined proofs of the presented results. The paper contains results of many authors as well as a preview of results of a forthcoming paper of Brendle, Guzmán, Hrušák, and Raghavan.

  • Corson reflections
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-13
    Ilijas Farah; Menachem Magidor

    A reflection principle for Corson compacta holds in the forcing extension obtained by Levy-collapsing a supercompact cardinal to ℵ2. In this model, a compact Hausdorff space is Corson if and only if all of its continuous images of weight ℵ1 are Corson compact. We use the Gelfand–Naimark duality, and our results are stated in terms of unital abelian C⁎-algebras.

  • An Efimov space with character less than s
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-13
    Alan Dow

    It is consistent that there is a compact space of character less than the splitting number in which there are no converging sequences. Such a space is an Efimov space.

  • Definable MAD families and forcing axioms
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-13
    Vera Fischer; David Schrittesser; Thilo Weinert

    We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π21 infinite MAD families” implies that ω1 is a remarkable cardinal in L. In other words, under BPFA and an anti-large cardinal assumption there is a Π21 infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ1-reflecting cardinal above

  • The Borel complexity of von Neumann equivalence
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-13
    Inessa Moroz; Asger Törnquist

    We prove that for a countable discrete group Γ containing a copy of the free group Fn, for some 2≤n≤∞, as a normal subgroup, the equivalence relations of conjugacy, orbit equivalence and von Neumann equivalence of the ergodic a.e. free probability measure preserving actions of Γ are analytic non-Borel equivalence relations in the Polish space of probability measure preserving Γ-actions. As a consequence

  • Two Applications of Topology to Model Theory
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-13
    Christopher J. Eagle; Clovis Hamel; Franklin D. Tall

    By utilizing the topological concept of pseudocompactness, we simplify and improve a proof of Caicedo, Dueñez, and Iovino concerning Terence Tao's metastability. We also pinpoint the exact relationship between the Omitting Types Theorem and the Baire Category Theorem by developing a machine that turns topological spaces into abstract logics.

  • Separating families and order dimension of Turing degrees
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-13
    Ashutosh Kumar; Dilip Raghavan

    We study families of functions and linear orders which separate countable subsets of the continuum from points. As an application, we show that the order dimension of the Turing degrees, denoted dimT, cannot be decided in ZFC. We also provide a combinatorial description of dimT and show that the Turing degrees have the largest order dimension among all locally countable partial orders of size continuum

  • Some remarks on the Open Coloring Axiom
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-12
    Justin Tatch Moore

    This note contains two results relating to the problem of whether the Open Coloring Axiom implies that the continuum is ℵ2. It also establishes that Farah's OCA∞ is equivalent to OCA.

  • Convergent sequences in topological groups
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-12
    Michael Hrušák; Alexander Shibakov

    We survey recent developments concerning the role of convergent sequences in topological groups. We present the Invariant Ideal Axiom and announce its effect on convergence properties in topological groups, in particular, the consistency of the fact that every countable sequential topological group is either metrizable or kω. We also outline a construction of a countably compact topological group without

  • The canonical pairs of bounded depth Frege systems
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-02
    Pavel Pudlák

    The canonical pair of a proof system P is the pair of disjoint NP sets where one set is the set of all satisfiable CNF formulas and the other is the set of CNF formulas that have P-proofs bounded by some polynomial. We give a combinatorial characterization of the canonical pairs of depth d Frege systems. Our characterization is based on certain games, introduced in this article, that are parametrized

  • Computable Irrational Numbers with Representations of Surprising Complexity
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-10-06
    Ivan Georgiev; Lars Kristiansen; Frank Stephan

    Cauchy sequences, Dedekind cuts, base-10 expansions and continued fractions are examples of well-known representations of irrational numbers. But there exist others, not so popular, which can be defined using various kinds of sum approximations and best approximations. In this paper we investigate the complexity of a number of such representations. For any fast-growing computable function f, we define

  • Stationary and closed rainbow subsets
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-09-23
    Shimon Garti; Jing Zhang

    We study the structural rainbow Ramsey theory at uncountable cardinals. Compared to the usual rainbow Ramsey theory, the variation focuses on finding a rainbow subset that not only is of a certain cardinality but also satisfies certain structural constraints, such as being stationary or closed in its supremum. In the process of dealing with cardinals greater than ω1, we uncover some connections between

  • Small models, large cardinals, and induced ideals
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-09-23
    Peter Holy; Philipp Lücke

    We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important

  • On the number of independent orders
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-09-15
    Kota Takeuchi; Akito Tsuboi

    We investigate the model-theoretic invariant κsrdm(T), which was introduced by Shelah, and prove that κsrdm(T) is sub-additive. An infinite value of κsrdm(T) leads to the equality κsrdm(T)=κsrd1(T). We apply the same proof method to analyze the other invariant κirdm(T) and show that it is also sub-additive, achieving an improvement of Shelah's result.

  • First-order model theory of free projective planes
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-09-22
    Tapani Hyttinen; Gianluca Paolini

    We prove that the theory of open projective planes is complete and strictly stable, and infer from this that Marshall Hall's free projective planes (πn:4⩽n⩽ω) are all elementary equivalent and that their common theory is strictly stable and decidable, being in fact the theory of open projective planes. We further characterize the elementary substructure relation in the class of open projective planes

  • Derivatives of normal functions in reverse mathematics
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-09-24
    Anton Freund; Michael Rathjen

    Consider a normal function f on the ordinals (i. e. a function f that is strictly increasing and continuous at limit stages). By enumerating the fixed points of f we obtain a faster normal function f′, called the derivative of f. The present paper investigates this important construction from the viewpoint of reverse mathematics. Within this framework we must restrict our attention to normal functions

  • Algebraic combinatorics in bounded induction
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-09-03
    Joaquín Borrego-Díaz

    In this paper, new methods for analyzing models of weak subsystems of Peano Arithmetic are proposed. The focus will be on the study of algebro-combinatoric properties of certain definable cuts. Their relationship with segments that satisfy more induction, with those limited by the standard powers/roots of an element, and also with definable sets in Bounded Induction is studied. As a consequence, some

  • Tukey order, calibres and the rationals
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-08-17
    Paul Gartside, Ana Mamatelashvili

    One partially ordered set, Q, is a Tukey quotient of another, P – denoted P≥TQ – if there is a map ϕ:P→Q carrying cofinal sets of P to cofinal sets of Q. Let X be a space and denote by K(X) the set of compact subsets of X, ordered by inclusion. For certain separable metrizable spaces M, Tukey upper and lower bounds of K(M) are calculated. Results on invariants of K(M)'s are deduced. The structure of

  • When Pκ(λ) (vaguely) resembles κ
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-08-14
    Pierre Matet

    Let μ<κ<λ be three infinite cardinals, the first two being regular. Assuming the existence of a cofinal subset of (Pκ(λ),⊆) of size λ, we define an ideal on Pκ(λ) which we argue to be the analog for Pκ(λ) of NSκ|Eμκ (the restriction of the nonstationary ideal on κ to the set of limit ordinals less than κ of cofinality μ). We show that this ideal on Pκ(λ) is the ideal dual to the well-known μ-club filter

  • Probabilistic Characterisation of Models of First-Order Theories
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-08-11
    Soroush Rafiee Rad

    We study probabilistic characterisation of a random model of a finite set of first order axioms. Given a set of first order axioms T and a structure M which we only know is a model of T, we are interested in the probability that M would satisfy a sentence ψ. Answering this question for all sentences in the language will give a probability distribution over the set of sentences which can be regarded

  • Ax-Schanuel and strong minimality for the j-function
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-08-04
    Vahagn Aslanyan

    Let K:=(K;+,⋅,D,0,1) be a differentially closed field of characteristic 0 with field of constants C. In the first part of the paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation E(x,y) and the geometry of the fibres Us:={y:E(s,y)∧y∉C} where s is a non-constant element. We show that certain types of predimension inequalities imply

  • Computable analogs of cardinal characteristics: Prediction and rearrangement
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-08-03
    Iván Ongay-Valverde, Paul Tveite

    There has recently been work by multiple groups in extracting the properties associated with cardinal invariants of the continuum and translating these properties into similar analogous combinatorial properties of computational oracles. Each property yields a highness notion in the Turing degrees. In this paper we study the highness notions that result from the translation of the evasion number and

  • Towards the entropy-limit conjecture
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-07-24
    Jürgen Landes, Soroush Rafiee Rad, Jon Williamson

    The maximum entropy principle is widely used to determine non-committal probabilities on a finite domain, subject to a set of constraints, but its application to continuous domains is notoriously problematic. This paper concerns an intermediate case, where the domain is a first-order predicate language. Two strategies have been put forward for applying the maximum entropy principle on such a domain:

  • Preservation theorems for Namba forcing
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-07-23
    Osvaldo Guzmán, Michael Hrušák, Jindřich Zapletal

    We study preservation properties of Namba forcing on κ. We prove that if I is an ideal with a Borel base on ωω and κ>ω1 is a regular cardinal less than the uniformity number or bigger than the covering number of I, then the κ-Namba forcing preserves the covering of I. The situation at κ=ω1, also treated here, is more complex.

  • Complexity of syntactical tree fragments of Independence-Friendly logic
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-07-17
    Fausto Barbero

    A dichotomy result of Sevenster (2014) [29] completely classified the quantifier prefixes of regular Independence-Friendly (IF) logic according to the patterns of quantifier dependence they contain. On one hand, prefixes that contain “Henkin” or “signalling” patterns were shown to characterize fragments of IF logic that capture NP-complete problems; all the remaining prefixes were shown instead to

  • Open core and small groups in dense pairs of topological structures
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-07-16
    Elías Baro, Amador Martin-Pizarro

    Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct

  • Finitely generated groups are universal among finitely generated structures
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-07-03
    Matthew Harrison-Trainor, Meng-Che “Turbo” Ho

    Universality has been an important concept in computable structure theory. A class C of structures is universal if, informally, for any structure of any kind there is a structure in C with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely

  • |˜-divisibility of ultrafilters
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-07-03
    Boris Šobot

    We further investigate a divisibility relation on the set βN of ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a hierarchy in which a position of every ultrafilter depends on the set of prime ultrafilters it is divisible by. We also construct ultrafilters with many immediate successors in this hierarchy and find positions

  • Filter-linkedness and its effect on preservation of cardinal characteristics
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-06-29
    Jörg Brendle, Miguel A. Cardona, Diego A. Mejía

    We introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “μ-F-linked” and “θ-F-Knaster” for posets in a natural way. We show that θ-F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct θ-Fr-Knaster

  • On equivalence relations generated by Cauchy sequences in countable metric spaces
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-06-25
    Longyun Ding, Kai Gu

    Let X be the set of all metrics on ω, and let Xcpt be the set of all metrics r on ω such that the completion of (ω,r) is compact. We define the Cauchy sequence equivalence relation Ecs on X as: rEcss iff the set of Cauchy sequences in (ω,r) is same as in (ω,s). We also denote Ecsc=Ecs↾Xcpt. We show that Ecs is a Π11-complete equivalence relation, while Ecsc is a Π30 equivalence relation. We also show

  • The tree property at double successors of singular cardinals of uncountable cofinality with infinite gaps
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-06-18
    Mohammad Golshani; Alejandro Poveda

    Assume that κ and λ are respectively strong and weakly compact cardinals with λ>κ. Fix Θ≥λ a cardinal with cof(Θ)>κ and cof(δ)=δ<κ. Assuming the GCH≥κ holds, we construct a generic extension of the universe where κ is a strong limit cardinal, cof(κ)=δ, 2κ=Θ and TP(κ++) holds. This extends the main result of [4] for uncountable cofinalities.

  • Proofs and surfaces
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-06-01
    Djordje Baralić, Pierre-Louis Curien, Marina Milićević, Jovana Obradović, Zoran Petrić, Mladen Zekić, Rade T. Živaljević

    A formal sequent system dealing with Menelaus' configurations is introduced in this paper. The axiomatic sequents of the system stem from 2-cycles of Δ-complexes. The Euclidean and projective interpretations of the sequents are defined and a soundness result is proved. This system is decidable and its provable sequents deliver incidence results. A cyclic operad structure tied to this system is presented

  • Structure and representation of semimodules over inclines
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-29
    Ruiqi Bai, Yichuan Yang

    An incline S is a commutative semiring where r+1=1 for any r∈S. We note that the ideal lattice of an S-semimodule is naturally an S-semimodule and so is its congruence lattice when S is transitive. We prove that the categories of complete S-semimodules, together with dual functor, internal hom and tensor product, is a ⋆-autonomous category. We define the locally and globally maximal congruences which

  • The FAN principle and weak König's lemma in herbrandized second-order arithmetic
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-28
    Fernando Ferreira

    We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically

  • Join-completions of partially ordered algebras
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-23
    José Gil-Férez, Luca Spada, Constantine Tsinakis, Hongjun Zhou

    We present a systematic study of join-extensions and join-completions of partially ordered algebras, which naturally leads to a refined and simplified treatment of fundamental results and constructions in the theory of ordered structures ranging from properties of the Dedekind–MacNeille completion to the proof of the finite embeddability property for a number of varieties of lattice-ordered algebras

  • Ultrafilters, finite coproducts and locally connected classifying toposes
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-19
    Richard Garner

    We prove a single category-theoretic result encapsulating the notions of ultrafilters, ultrapower, ultraproduct, tensor product of ultrafilters, the Rudin–Kiesler partial ordering on ultrafilters, and Blass's category of ultrafilters UF. The result in its most basic form states that the category FC(Set,Set) of finite-coproduct-preserving endofunctors of Set is equivalent to the presheaf category [UF

  • Polish metric spaces with fixed distance set
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-19
    Riccardo Camerlo, Alberto Marcone, Luca Motto Ros

    We study Polish spaces for which a set of possible distances A⊆R+ is fixed in advance. We determine, depending on the properties of A, the complexity of the collection of all Polish metric spaces with distances in A, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that A must have in order that all Polish spaces with

  • Diagonal supercompact Radin forcing
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-18
    Omer Ben-Neria, Chris Lambie-Hanson, Spencer Unger

    Motivated by the goal of constructing a model in which there are no κ-Aronszajn trees for any regular κ>ℵ1, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square fail.

  • Rules with parameters in modal logic II
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-15
    Emil Jeřábek

    We analyze the computational complexity of admissibility and unifiability with parameters in transitive modal logics. The class of cluster-extensible (clx) logics was introduced in the first part of this series of papers [8]. We completely classify the complexity of unifiability or inadmissibility in any clx logic as being complete for one of Σ2exp, NEXP, coNEXP, PSPACE, or Π2p. In addition to the

  • Modal extension of ideal paraconsistent four-valued logic and its subsystem
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-15
    Norihiro Kamide, Yoni Zohar

    This study aims to introduce a modal extension M4CC of Arieli, Avron, and Zamansky's ideal paraconsistent four-valued logic 4CC as a Gentzen-type sequent calculus and prove the Kripke-completeness and cut-elimination theorems for M4CC. The logic M4CC is also shown to be decidable and embeddable into the normal modal logic S4. Furthermore, a subsystem of M4CC, which has some characteristic properties

  • Epimorphism surjectivity in varieties of Heyting algebras
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-13
    T. Moraschini, J.J. Wannenburg

    It was shown recently that epimorphisms need not be surjective in a variety K of Heyting algebras, but only one counter-example was exhibited in the literature until now. Here, a continuum of such examples is identified, viz. the variety generated by the Rieger-Nishimura lattice, and all of its (locally finite) subvarieties that contain the original counter-example K. It is known that, whenever a variety

  • Perfect tree forcings for singular cardinals
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-13
    Natasha Dobrinen, Dan Hathaway, Karel Prikry

    We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals 〈κn:n<ω〉, Prikry defined the forcing P of all perfect subtrees of ∏n<ωκn, and proved that for κ=supn<ω⁡κn, assuming the necessary cardinal arithmetic, the Boolean completion

  • A premouse inheriting strong cardinals from V
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-12
    Farmer Schlutzenberg

    We identify a premouse inner model L[E], such that for any coarsely iterable background universe R modelling ZFC, L[E]R is a proper class premouse of R inheriting all strong and Woodin cardinals from R. For each ordinal α, L[E]R|α is (ω,α)-iterable, via iteration trees which lift to coarse iteration trees on R. We prove that (k+1)-condensation follows from (k+1)-solidity and (k,ω1+1)-iterability (that

  • Silver type theorems for collapses
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-11
    Moti Gitik

    Let κ be a cardinal of cofinality ω1 witnessed by a club of cardinals 〈κα|α<ω1〉. We study Silver's type effects of collapsing of κα+'s on κ+. A model in which κα+'s (and also κ+) are collapsed on a stationary co-stationary set is constructed.

  • Some constructions of ultrafilters over a measurable cardinal
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-06
    Moti Gitik

    Some non-normal κ-complete ultrafilters over a measurable κ with special properties are constructed. Questions by A. Kanamori [4] about infinite Rudin-Frolik sequences, discreteness and products are answered.

  • Algebraically closed structures in positive logic
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-06
    Mohammed Belkasmi

    In this paper we extend of the notion of algebraically closed given in the case of groups and skew fields to an arbitrary h-inductive theory. The main subject of this paper is the study of the notion of positive algebraic closedness and its relationship with the notion of positive closedness and the amalgamation property.

  • Computability of pseudo-cubes
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-05-04
    Marko Horvat, Zvonko Iljazović, Bojan Pažek

    We examine topological pairs (Δ,Σ) which have computable type: if X is a computable topological space and f:Δ→X a topological embedding such that f(Δ) and f(Σ) are semicomputable sets in X, then f(Δ) is a computable set in X. It is known that (D,W) has computable type, where D is the Warsaw disc and W is the Warsaw circle. In this paper we identify a class of topological pairs which are similar to

  • A forcing axiom for a non-special Aronszajn tree
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-04-30
    John Krueger

    Suppose that T⁎ is an ω1-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA(T⁎) for proper forcings which preserve these properties of T⁎. We prove that PFA(T⁎) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than ω1, and the P-ideal dichotomy. On the other

  • The theory of ceers computes true arithmetic
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-04-10
    Uri Andrews, Noah Schweber, Andrea Sorbi

    We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the structure comprised of the light ceers. We also show the same for the structure of I-degrees in the dark, light, or complete structure. In each case, we show that

  • A classification of the cofinal structures of precompacta
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-04-08
    Aviv Eshed, M. Vicenta Ferrer, Salvador Hernández, Piotr Szewczak, Boaz Tsaban

    We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups C(X,R) of continuous real-valued functions on complete metric spaces X, with respect to the compact-open topology

  • On expansions of (Z,+,0)
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-03-25
    Quentin Lambotte, Françoise Point

    Call a (strictly increasing) sequence (rn) of natural numbers regular if it satisfies the following condition: rn+1/rn→θ∈R>1∪{∞} and, if θ is algebraic, then (rn) satisfies a linear recurrence relation whose characteristic polynomial is the minimal polynomial of θ. Our main result states that (Z,+,0,R) is superstable whenever R is enumerated by a regular sequence. We give two proofs of this result

  • Expansions of real closed fields that introduce no new smooth functions
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-03-23
    Pantelis E. Eleftheriou, Alex Savatovsky

    We prove the following theorem: let R˜ be an expansion of the real field R‾, such that every definable set (I) is a uniform countable union of semialgebraic sets, and (II) contains a “semialgebraic chunk”. Then every definable smooth function f:X⊆Rn→R with open semialgebraic domain is semialgebraic. Conditions (I) and (II) hold for various d-minimal expansions R˜=〈R‾,P〉 of the real field, such as when

  • The density zero ideal and the splitting number
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-03-19
    Dilip Raghavan

    The main result of this paper is an improvement of the upper bound on the cardinal invariant cov⁎(Z0) that was discovered in [11]. Here Z0 is the ideal of subsets of the set of natural numbers that have asymptotic density zero. This improved upper bound is also dualized to get a better lower bound on the cardinal non⁎(Z0). En route some variations on the splitting number are introduced and several

  • Some lower bounds on Shelah rank in the free group
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-03-12
    Javier de la Nuez González, Chloé Perin, Rizos Sklinos

    We give some lower bounds on the Shelah rank of varieties in the free group whose coordinate groups are hyperbolic towers.

  • M-separable spaces of functions are productive in the Miller model
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-03-10
    Dušan Repovš, Lyubomyr Zdomskyy

    We prove that in the Miller model, every M-separable space of the form Cp(X), where X is metrizable and separable, is productively M-separable, i.e., Cp(X)×Y is M-separable for every countable M-separable Y.

  • Definable groups in models of Presburger Arithmetic
    Ann. Pure Appl. Logic (IF 0.752) Pub Date : 2020-03-09
    Alf Onshuus, Mariana Vicaría

    This paper is devoted to understand groups definable in Presburger Arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem 2. Every bounded abelian group definable in a model (Z,+,<) of Presburger Arithmetic is definably isomorphic to (Z,+)n mod out by a lattice.

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